Theorem isnumbasgrplem2 42670

Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)

Assertion

Ref	Expression
isnumbasgrplem2	⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 17187	. . 3 ⊢ Base Fn V
2		ssv 4001	. . 3 ⊢ Grp ⊆ V
3		fvelimab 6970	. . 3 ⊢ ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
4	1, 2, 3	mp2an 690	. 2 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5		harcl 9584	. . . . . 6 ⊢ (har‘𝑆) ∈ On
6		onenon 9974	. . . . . 6 ⊢ ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
7	5, 6	ax-mp 5	. . . . 5 ⊢ (har‘𝑆) ∈ dom card
8		xpnum 9976	. . . . 5 ⊢ (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
9	7, 7, 8	mp2an 690	. . . 4 ⊢ ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10		ssun1 4170	. . . . . . . 8 ⊢ 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11		simpr 483	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
12	10, 11	sseqtrrid 4030	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13		fvex 6909	. . . . . . . 8 ⊢ (Base‘𝑥) ∈ V
14	13	ssex 5322	. . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
15	12, 14	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
16	7	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17		simp1l 1194	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18	12	3ad2ant1 1130	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19		simp2 1134	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3977	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21		ssun2 4171	. . . . . . . . . . 11 ⊢ (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
22	21, 11	sseqtrrid 4030	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23	22	3ad2ant1 1130	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24		simp3 1135	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
25	23, 24	sseldd 3977	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26		eqid 2725	. . . . . . . . 9 ⊢ (Base‘𝑥) = (Base‘𝑥)
27		eqid 2725	. . . . . . . . 9 ⊢ (+g‘𝑥) = (+g‘𝑥)
28	26, 27	grpcl 18906	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
29	17, 20, 25, 28	syl3anc 1368	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
30		simp1r 1195	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
31	29, 30	eleqtrd 2827	. . . . . 6 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32		simplll 773	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
33	22	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34		simprl 769	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
35	33, 34	sseldd 3977	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36		simprr 771	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
37	33, 36	sseldd 3977	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
38	12	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39		simplr 767	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ 𝑆)
40	38, 39	sseldd 3977	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
41	26, 27	grplcan 18965	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
42	32, 35, 37, 40, 41	syl13anc 1369	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
43		simplll 773	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑥 ∈ Grp)
44	12	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45		simprr 771	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ 𝑆)
46	44, 45	sseldd 3977	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ (Base‘𝑥))
47		simprl 769	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
48	44, 47	sseldd 3977	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ (Base‘𝑥))
49	22	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50		simplr 767	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (har‘𝑆))
51	49, 50	sseldd 3977	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (Base‘𝑥))
52	26, 27	grprcan 18938	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
53	43, 46, 48, 51, 52	syl13anc 1369	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
54		harndom 9587	. . . . . . 7 ⊢ ¬ (har‘𝑆) ≼ 𝑆
55	54	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
56	15, 16, 16, 31, 42, 53, 55	unxpwdom3 42661	. . . . 5 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼* ((har‘𝑆) × (har‘𝑆)))
57		wdomnumr 10089	. . . . . 6 ⊢ (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
58	9, 57	ax-mp 5	. . . . 5 ⊢ (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
59	56, 58	sylib 217	. . . 4 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60		numdom 10063	. . . 4 ⊢ ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
61	9, 59, 60	sylancr 585	. . 3 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
62	61	rexlimiva 3136	. 2 ⊢ (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
63	4, 62	sylbi 216	1 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∃wrex 3059  Vcvv 3461   ∪ cun 3942   ⊆ wss 3944   class class class wbr 5149   × cxp 5676  dom cdm 5678   “ cima 5681  Oncon0 6371   Fn wfn 6544  ‘cfv 6549  (class class class)co 7419   ≼ cdom 8962  harchar 9581   ≼^* cwdom 9589  cardccrd 9960  Basecbs 17183  +_gcplusg 17236  Grpcgrp 18898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-1cn 11198  ax-addcl 11200

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-oi 9535  df-har 9582  df-wdom 9590  df-card 9964  df-acn 9967  df-nn 12246  df-slot 17154  df-ndx 17166  df-base 17184  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-minusg 18902

This theorem is referenced by:  isnumbasabl  42672  isnumbasgrp  42673